A079165 -id:A079165 - cp's OEIS frontend

A160327 Decimal expansion of (e-1)/(e+1).

4, 6, 2, 1, 1, 7, 1, 5, 7, 2, 6, 0, 0, 0, 9, 7, 5, 8, 5, 0, 2, 3, 1, 8, 4, 8, 3, 6, 4, 3, 6, 7, 2, 5, 4, 8, 7, 3, 0, 2, 8, 9, 2, 8, 0, 3, 3, 0, 1, 1, 3, 0, 3, 8, 5, 5, 2, 7, 3, 1, 8, 1, 5, 8, 3, 8, 0, 8, 0, 9, 0, 6, 1, 4, 0, 4, 0, 9, 2, 7, 8, 7, 7, 4, 9, 4, 9, 0, 6, 4, 1, 5, 1, 9, 6, 2, 4, 9, 0, 5, 8, 4, 3, 4, 8

Offset: 0

Harry J. Smith, May 09 2009

			0.462117157260009758502318483643672548730289280330113038552731815838080...

Harry J. Smith, Table of n, a(n) for n = 0..20000
Index entries for transcendental numbers

Cf. A016825 (continued fraction), A086403/A079165 (convergents).

Cf. A000367, A002445.

SetDefaultRealField(RealField(100)); (Exp(1) - 1)/(Exp(1) + 1); // G. C. Greubel, Oct 05 2018

RealDigits[(E-1)/(E+1), 10, 100][[1]] (* G. C. Greubel, Oct 05 2018 *)

default(realprecision, 20080); x=tanh(1/2)*10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b160327.txt", n, " ", d));

(exp(1)-1)/(exp(1)+1) \\ Altug Alkan, Oct 05 2018

(e-1)/(e+1) = tanh(1/2).
Equals 2 * Sum_{k>=1} (2^(2*k)-1)*B(2*k)/(2*k)!, where B(2*k) = A000367(k)/A002445(k) are the Bernoulli numbers. - Amiram Eldar, Nov 25 2020
Equals -i * tan(i/2). - Michal Paulovic, Jan 03 2023

A086403 Numerators in continued fraction representation of (e-1)/(e+1).

Original entry on oeis.org

1, 6, 61, 860, 15541, 342762, 8927353, 268163352, 9126481321, 347074453550, 14586253530421, 671314736852916, 33580323096176221, 1814008761930368850, 105246088515057569521, 6527071496695499679152, 430891964870418036393553, 30168964612425958047227862

Offset: 1

Gary W. Adamson, Jul 18 2003

nonn

			a(4) = 860 = closest integer to[(e-1)/(e+1)]*A079165(4); = floor(860.0000292...) = 860. 860/1861 = [2, 6, 10, 14] = .462117141...; (e-1)/(e+1) = .462117157...

Calvin C. Clawson, "Mathematical Mysteries", Perseus, 1999, p. 225.

Alois P. Heinz, Table of n, a(n) for n = 1..360

Cf. A000108, A016825, A033184, A079165.

b:= proc(n) local i, q;
      q:= 0;
      for i to n do
        q:= 1/(q+4*(n-i)+2)
      od; q
    end:
a:= n-> numer(b(n)):
seq(a(n), n=1..20);  # Alois P. Heinz, Feb 03 2012
numtheory:-cfrac((exp(1)-1)/(exp(1)+1),50,'convergents'):
map(numer,convergents[2..-2]); # Robert Israel, Apr 26 2016

Numerator@ FromContinuedFraction@ ContinuedFraction[(E - 1)/(E + 1), #] & /@ Range[2, 19] (* Michael De Vlieger, Apr 26 2016 *)

a(n):=(sum((2*n-2*k+1)!/((2*k-1)!*(n-2*k+1)!),k,1,(n+1)/2));
taylor(sinh((1-sqrt(1-4*x))/2)/sqrt(1-4*x),x,0,10); /* Vladimir Kruchinin, Apr 26 2016 */

Partial quotients in continued fraction representation of (e-1)/(e+1) are A016825: [2, 6, 10, 14, 18...], the convergents being: [2] = 1/2, [2, 6] = 6/13, [2, 6, 10] = 61/132...etc.; denominators are A079165 starting with n=1: 2, 13, 132, 1861, 33630, 741721, 19318376... 2. a(n) = closest integer to [(e-1)/(e+1)]*A079165(n), n>0
E.g.f.: sinh((1-sqrt(1-4*x))/2)/sqrt(1-4*x). - Vladimir Kruchinin, Apr 26 2016
a(n) = Sum_{k=1..(n+1)/2} (2*n-2*k+1)!/((2*k-1)!*(n-2*k+1)!). - Vladimir Kruchinin, Apr 26 2016
a(n) = -((-1)^n*sqrt(Pi/exp(1))*BesselI((2*n+1)/2, 1/2))/2 + (BesselK((2*n+1)/2, 1/2)*sinh(1/2))/sqrt(Pi), where BesselI(n,x) is the modified Bessel function of the first kind, BesselK(n,x) is the modified Bessel function of the second kind. - Ilya Gutkovskiy, Apr 26 2016
From Vaclav Kotesovec, Apr 27 2016: (Start)
a(n)/n! ~ BesselI(1/2, 1/2) * 2^(2*n-1)  / sqrt(n).
a(n) ~ sinh(1/2) * 2^(2*n + 1/2) * n^n  / exp(n).
(End)

More terms from Alois P. Heinz, Feb 03 2012

A079166 Square array read by antidiagonals of T(n,k)=(4k-2)*T(n,k-1)+T(n,k-2) with T(n,0)=1 and T(n,1)=n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 10, 7, 2, 1, 141, 71, 13, 3, 1, 2548, 1001, 132, 19, 4, 1, 56197, 18089, 1861, 193, 25, 5, 1, 1463670, 398959, 33630, 2721, 254, 31, 6, 1, 43966297, 10391023, 741721, 49171, 3581, 315, 37, 7, 1, 1496317768, 312129649, 19318376, 1084483

Offset: 0

Henry Bottomley, Dec 31 2002

			Rows start: 1,0,1,10,141,2548,56197,...; 1,1,7,71,1001,18089,398959,...; 1,2,13,132,1861,33630,741721,...; 1,3,19,193,2721,49171,1084483,...; etc.

Rows include A002119 (unsigned), A079165, A001517. Columns include A000012, A001477, A016921 (offset).

T(n, k) =2*T(n-1, k)-T(n-2, k). T(n, k)/T(1, k) tends to ( (n-1)*e - (n-3) )/2 as k increases: e.g. T(3, k)/T(1, k) tends to e.

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A160327 Decimal expansion of (e-1)/(e+1).

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A086403 Numerators in continued fraction representation of (e-1)/(e+1).

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A079166 Square array read by antidiagonals of T(n,k)=(4k-2)*T(n,k-1)+T(n,k-2) with T(n,0)=1 and T(n,1)=n.

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